Collaborative Effort in Wall Painting: An Optimal Time Calculation

In this article, we will explore a practical example of collaborative work and time calculation in the context of painting a wall. Specifically, we are looking into the scenario where Brian can paint a wall red in 12 days, while Kate can paint the same wall in 21 days. When they work on this task alternately, how much time would it take for them to complete the entire work?

Efficiency of Brian and Kate

The first step in solving this problem is to understand the daily work efficiency of Brian and Kate individually. Brian can complete the job in 12 days, thus completing 1/12 of the wall per day. Similarly, Kate, with her efficiency, can paint 1/21 of the wall per day.

Alternating Work and Efficiency Calculation

When they work alternately, in a span of 2 days, they would complete a combined effort of 1/12 1/21 11/84 of the wall. This efficiency can be explained mathematically; on the first day, Brian paints 1/12 of the wall, and on the second day, Kate paints 1/21 of the wall. Together, they complete 11/84 of the wall in a span of 2 days.

Calculation and Iteration

By repeating this process, we find that in 7 iterations (or 14 days), they would complete 77/84 of the wall. This is calculated as follows:

7 iterations × 11/84 77/84

Thus, the remaining work, which amounts to 7/84 of the wall, is equivalent to 1/12 of the wall.

Given that Brian can complete 1/12 of the wall in one day, the remaining work can be completed by Brian in just 1 day. Hence, the total time to complete the entire wall will be 14 days (for 77/84 of the work) 1 day (for the remaining 1/12) 15 days.

LCM Approach for Further Understanding

To provide a deeper insight, let's consider the total work as 84 units. This is the least common multiple (LCM) of 12 and 21, allowing us to convert their work rates into a common unit. Brian can paint 7 units (84/12) of the wall per day, while Kate can paint 4 units (84/21) per day. In 2 days, together they can paint 11 units (7 4) of the wall.

By the end of 14 days (7 iterations), they can paint 77 units (7 × 11) of the wall. The remaining work is 7 units, which Brian alone can complete in 1 day. Thus, the combined effort over 15 days perfectly completes the work.

Conclusion

In this collaborative project, even though the work is challenging, understanding the collaborative efficiency and alternating work methods helps in calculating the optimal time to complete the task. The total time required to paint the entire wall alternately by Brian and Kate is 15 days.

Key takeaways include understanding collaborative work, time calculations, and the efficiency of each individual in completing the task. This example provides a clear and practical approach to solving such real-world problems in a collaborative setting.